On a problem of Erdős

Let s≥2 be an integer. Denote by f ₁(s) the least integer so that every integer l>f ₁(s) is the sum of s distinct primes. Erd?s proved that f ₁(s)<p ₁+p ₂+?+p _s +Cslogs, where p _i is the ith prime and C is an absolute constant. In this paper, we prove that f ₁(s)=p ₁+p ₂+?+p _s +(1+o(1))slogs=p ₂+p ₃+?+p _s+1+o(slogs). This answers a question posed by P. Erd?s.     

On a problem of Erdős and Rado

We give some improved estimates for the digraph Ramsey numbersr(K _n ^* ,L _m ), the smallest numberp such that any digraph of orderp either has an independent set ofn vertices or contains a transitive tournament of orderm. By results of Baumgartner and of Erdős and Rado, this is equivalent to the following infinite partition problem: for an infinite cardinal κ and positive integersn andm, find the smallest numberp such that

On a problem of Erdős and Graham

In this paper we provide bounds for the size of the solutions of the Diophantine equation

$$\begin{aligned} x(x+1)(x+2)(x+3)(x+m)(x+m+1)(x+m+2)(x+m+3)=y^2, \end{aligned}$$

where \(4\le m\in \mathbb {N}\) is a parameter. We also determine all integral solutions for \(1\le m\le 10^6.\)

     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.     

On a Question of Erdős and Ulam

Ulam asked in 1945 if there is an everywhere dense rational set, i.e., 1 a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algebraic curves by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.     

On a bottleneck bipartition conjecture of Erdős

For a graphG, let (U,V)=max{e(U), e(V)} for a bipartition (U, V) ofV(G) withUV=V(G),UV=Ø. Define (G)=min_(U,V ){(U,V)}. Paul Erds conjectures . This paper verifies the conjecture and shows .This work was part of the author's Ph. D. thesis at the University of New Mexico. Research Partially supported by NSA Grant MDA904-92-H-3050.     

Variants of the Erdős–Szekeres and Erdős–Hajnal Ramsey problems

Given integers $\ell ,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell }$ with vertex set $v_1<v_2<\cdots <v_n$ and all edges of the form $v_i{v}_j$ where $|i-j|\le \ell$. The Ramsey number $r(P_n^{\ell },P_n^{\ell })$ is the minimum $N$ such that every 2-coloring of $\left(\genfrac{}{}{0ex}{}{\left[N\right]}{2}\right)$ results in a monochromatic copy of $P_n^{\ell }$. It is well-known that $r(P_n^1,P_n^1)={(n-1)}^2+1$. For $\ell >1$, Balko–Cibulka–Král–Kynčl proved that $r(P_n^{\ell },P_n^{\ell })<c_{\ell }{n}^{128\ell }$ and asked for the growth rate for fixed $\ell$. When $\ell =2$, we improve this upper bound substantially by proving $r(P_n^2,P_n^2)<c{n}^{19.5}$. Using this result, we determine the correct tower growth rate of the $k$-uniform hypergraph Ramsey number of a $(k+1)$-clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erdős–Hajnal hypergraph Ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.     

On the Erdös–Lax inequality concerning polynomials

Let $P(z)$ be a polynomial of degree n and for any complex number α, let $D_{\alpha }P(z):=nP(z)+(\alpha ?z)P^{\prime }(z)$ denote the polar derivative of $P(z)$ with respect to α. In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdöx–Lax theorem.     

On Erdős-Rado numbers

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by Erd?s and Rado, are given. These yield improvements over the known bounds for the arising Erd?s-Rado numbersER(k; l), where the numbersER(k; l) are defined as the least positive integern such that for every partition of thek-element subsets of a totally orderedn-element setX into an arbitrary number of classes there exists anl-element subsetY ofX, such that the set ofk-element subsets ofY is partitioned canonically (in the sense of Erd?s and Rado). In particular, it is shown that $$2^{c1} .l^2 \leqslant ER(2;l) \leqslant 2^{c_2 .l^2 .\log l} $$ for every positive integerl≥3, wherec ₁,c ₂ are positive constants. Moreover, new bounds, lower and upper, for the numbersER(k; l) for arbitrary positive integersk, l are given.     

Tauberian Theorem of Erdős Revisited

Dedicated to the memory of Paul Erdős In connection with the elementary proof of the prime number theorem, Erdős obtained a striking quadratic Tauberian theorem for sequences. Somewhat later, Siegel indicated in a letter how a powerful "fundamental relation" could be used to simplify the difficult combinatorial proof. Here the author presents his version of the (unpublished) Erdős–Siegel proof. Related Tauberian results by the author are described. Received December 20, 1999     

On Jensen-McShane’s inequality

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.     

On Hasse’s inequality

We give an elementary exposition of the little known work of Harold Davenport related to Hasse’s inequality. We formulate a new conjecture suggested by this proof that has implications for the classical Riemann hypothesis.     

On Jordan’s inequality

We present sharp upper and lower bounds for the function \(\sin (x)/x\). Our bounds are polynomials of degree 2n, where n is any nonnegative integer.     

On Bohr’s inequality

It is established that H. Bohr’s inequality \(\sum\nolimits_{k = 0}^\infty {\left| {{{f^{\left( k \right)} \left( 0 \right)} \mathord{\left/ {\vphantom {{f^{\left( k \right)} \left( 0 \right)} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right| \leqslant \sqrt 2 \left\| f \right\|_\infty }\) is sharp on the class H _∞.     

On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C _k denote a cycle of length k, and let C _k denote the set of cycles C _?, where 3≤?≤k and ? and k have the same parity. Erd?s and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C _k ) ～ z(n,F) — here we write f(n) ～ g(n) for functions f,g: ? → ? if lim _n→∞ f(n)/g(n)=1. They proved this when F ={C ₄} by showing that ex(n,{C ₄;C ₅})～z(n,C ₄). In this paper, we extend this result by showing that if ?∈{2,3,5} and k>2? is odd, then ex(n,C _2? ∪{C _k }) ～ z(n,C _2? ). Furthermore, if k>2?+2 is odd, then for infinitely many n we show that the extremal C _2? ∪{C _k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2?, and furthermore the asymptotic result does not hold when (?,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.